Can immunity from one disease give immunity to another?

Your friend is right to a degree. Studies have been done which show that children now are more prone to allergies and it appears to be because of over fastidious mums. It does not hurt for a child to have dirty play and to crawl around being exposed to general germs. It builds up immunity generally which makes the body dtronger when it comes to a "nasty" germ. It does not actually give immunity against another disease, but the body's response to invaders has been practiced and swings into action faster and more efficently.